What is the simplified value of [tan²(90° − θ) − sin²(90° − θ)] · cosec²(90° − θ) · cot²(90° − θ)?

Introduction / Context:
This problem asks you to simplify a trigonometric expression that uses complementary angles (90° − θ). It involves tan, sin, cosec and cot evaluated at 90° − θ. The question tests your understanding of complementary angle identities and how to simplify squared trigonometric expressions.

Given Data / Assumptions:
- Expression: [tan²(90° − θ) − sin²(90° − θ)] · cosec²(90° − θ) · cot²(90° − θ).
- θ is an angle such that the trigonometric functions involved are defined.

Concept / Approach:
We use complementary angle relationships: tan(90° − θ) = cot θ, sin(90° − θ) = cos θ, cosec(90° − θ) = sec θ, and cot(90° − θ) = tan θ. Substituting these into the expression converts everything into functions of θ, which can then be simplified using known identities like sec²θ = 1 + tan²θ and sin²θ + cos²θ = 1.

Step-by-Step Solution:
Step 1: Use tan(90° − θ) = cot θ, so tan²(90° − θ) = cot²θ.Step 2: Use sin(90° − θ) = cos θ, so sin²(90° − θ) = cos²θ.Step 3: Use cosec(90° − θ) = sec θ, so cosec²(90° − θ) = sec²θ.Step 4: Use cot(90° − θ) = tan θ, so cot²(90° − θ) = tan²θ.Step 5: Substitute into the expression to get [cot²θ − cos²θ] · sec²θ · tan²θ.Step 6: Rewrite cot²θ as cos²θ / sin²θ and sec²θ as 1 / cos²θ.Step 7: Then [cot²θ − cos²θ] = [cos²θ / sin²θ − cos²θ] = cos²θ(1 / sin²θ − 1) = cos²θ[(1 − sin²θ) / sin²θ] = cos²θ(cos²θ / sin²θ) = cos⁴θ / sin²θ.Step 8: Now multiply by sec²θ tan²θ: (cos⁴θ / sin²θ) * (1 / cos²θ) * (sin²θ / cos²θ) = (cos⁴θ / sin²θ) * (sin²θ / cos⁴θ) = 1.

Verification / Alternative check:
Choose a value such as θ = 30°. Evaluate each part numerically and compute the product. The result will be 1, confirming the algebraic simplification. Trying a few different angles will give the same result as long as the functions are defined.

Why Other Options Are Wrong:
The values 0, −1 and 2 do not agree with the exact simplification. Zero would suggest a factor vanished, which does not happen. A result of −1 or 2 usually comes from sign errors, incorrect complementary angle identities or mistakes in simplifying the squares of the functions.

Common Pitfalls:
A typical mistake is to confuse sec with cosec or tan with cot when applying complementary angle formulas. Another common error is to forget that sec²θ = 1 + tan²θ and instead misapply Pythagorean identities. Ensuring each substitution and simplification step is done carefully avoids these problems.

Final Answer:
The simplified value of the expression is 1.